imported>Dmitrii Kouznetsov |
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| == Summary == | | == Summary == |
| {{Image_Details|user
| | Importing file |
| |description = [[Complex map]] of [[tetration]] to base <math>b=1.52598338517+0.0178411853321\, \mathrm i</math>
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| is shown with lines of constant <math>u</math>
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| and lines of constant <math>v</math>, while
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| <math>
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| u+\mathrm i v= \mathrm{tet}_b(x\!+\!\mathrm i y)
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| </math>
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| |author = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
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| |date-created = 2014 August
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| |pub-country = Japan
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| |notes = I use this image in the article
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| D.Kouznetsov. Holomorphic ackermanns. 2015, in preparation.
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| |versions = http://mizugadro.mydns.jp/t/index.php/File:Ack4c.jpg
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| }}
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| == Licensing ==
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| {{CC|by|3.0}}
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| | |
| ==[[C++]] Generator of map==
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| Files
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| [[ado.cin]],
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| [[conto.cin]],
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| [[filog.cin]],
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| [[TetSheldonIma.inc]],
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| [[GLxw2048.inc]]
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| should be loaded to the working directory in order to compile the code below.
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| | |
| <nowiki>
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| #include <math.h>
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| #include <stdio.h>
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| #include <stdlib.h>
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| #define DB double
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| #define DO(x,y) for(x=0;x<y;x++)
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| #include <complex>
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| typedef std::complex<double> z_type;
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| #define Re(x) x.real()
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| #define Im(x) x.imag()
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| #define I z_type(0.,1.)
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| #include "conto.cin"
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| #include "filog.cin"
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| | |
| z_type b=z_type( 1.5259833851700000, 0.0178411853321000);
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| z_type a=log(b);
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| z_type Zo=Filog(a);
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| z_type Zc=conj(Filog(conj(a)));
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| DB A=32.;
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| z_type tetb(z_type z){ int k; DB t; z_type c, cu,cd;
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| #include "GLxw2048.inc"
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| int K=2048;
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| //#include "ima6.inc"
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| #include "TetSheldonIma.inc"
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| z_type E[2048],G[2048];
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| DO(k,K){c=F[k]; E[k]=log(c)/a; G[k]=exp(a*c);}
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| c=0.;
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| z+=z_type(0.1196573712872846, 0.1299776198056910);
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| DO(k,K){t=A*GLx[k];c+=GLw[k]*(G[k]/(z_type( 1.,t)-z)-E[k]/(z_type(-1.,t)-z));}
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| cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) );
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| cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) );
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| c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd;
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| return c;}
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| | |
| int main(){ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d;
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| int M=601,M1=M+1;
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| int N=461,N1=N+1;
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| | |
| DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
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| char v[M1*N1]; // v is working array
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| FILE *o;o=fopen("tetsheldonmap.eps","w");ado(o,602,202);
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| fprintf(o,"301 101 translate\n 10 10 scale\n");
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| DO(m,M1)X[m]=-30.+.1*(m);
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| DO(n,200)Y[n]=-10.+.05*n;
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| Y[200]=-.01;
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| Y[201]= .01;
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| for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-1.);
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| for(m=-30;m<31;m++){if(m==0){M(m,-10.2)L(m,10.2)} else{M(m,-10)L(m,10)}}
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| for(n=-10;n<11;n++){ M( -30,n)L(30,n)}
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| fprintf(o,".008 W 0 0 0 RGB S\n");
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| DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
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| DO(n,N1){y=Y[n];
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| for(m=295;m<305;m++)
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| {x=X[m]; //printf("%5.2f\n",x);
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| z=z_type(x,y);
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| c=tetb(z);
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| p=Re(c);q=Im(c);
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| if(p>-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
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| d=c;
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| for(k=1;k<31;k++)
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| { m1=m+k*10; if(m1>M) break;
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| d=exp(a*d);
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| p=Re(d);q=Im(d);
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| if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
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| }
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| d=c;
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| for(k=1;k<31;k++)
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| { m1=m-k*10; if(m1<0) break;
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| d=log(d)/a;
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| p=Re(d);q=Im(d);
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| if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
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| }
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| }}
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| | |
| fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1;q=.5;
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| for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".02 W 0 .6 0 RGB S\n");
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| for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".02 W .9 0 0 RGB S\n");
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| for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".02 W 0 0 .9 RGB S\n");
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| for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".08 W .9 0 0 RGB S\n");
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| for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 .9 RGB S\n");
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| conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".08 W .6 0 .6 RGB S\n");
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| for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 0 RGB S\n");
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| fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
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| system("epstopdf tetsheldonmap.eps");
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| system( "open tetsheldonmap.pdf");
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| getchar(); system("killall Preview");
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| }
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| </nowiki>
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| | |
| ==[[Latex]] generator of labels==
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| | |
| <nowiki>
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| \documentclass[12pt]{article}
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| \paperwidth 640px
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| \paperheight 1006px
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| \textwidth 1394px
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| \textheight 1300px
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| \topmargin -104px
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| \oddsidemargin -92px
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| \usepackage{graphics}
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| \usepackage{rotating}
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| \newcommand \sx {\scalebox}
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| \newcommand \rot {\begin{rotate}}
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| \newcommand \ero {\end{rotate}}
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| \newcommand \ing {\includegraphics}
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| \newcommand \rmi {\mathrm{i}}
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| \begin{document}
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| {\begin{picture}(608,1006) %\put(12,0){\ing{penma}}
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| \put(0,0){\ing{acker2}}
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| \put(281,988){\sx{3.}{$y$}}
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| \put(279,895){\sx{3.}{$6$}}
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| \put(279,795){\sx{3.}{$5$}}
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| \put(279,694){\sx{3.}{$4$}}
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| \put(279,594){\sx{3.}{$3$}}
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| \put(279,468){\sx{3.}{$\mathrm e$}}
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| \put(279,494){\sx{3.}{$2$}}
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| \put(279,394){\sx{3.}{$1$}}
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| \put(279,294){\sx{3.}{$0$}}
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| \put(258,193){\sx{3.}{$-1$}}
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| \put(258, 93){\sx{3.}{$-2$}}
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| \put( 80,274){\sx{3.}{$-2$}}
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| \put(180,274){\sx{3.}{$-1$}}
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| \put(296,274){\sx{3.}{$0$}}
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| \put(396,274){\sx{3.}{$1$}}
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| \put(496,274){\sx{3.}{$2$}}
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| \put(586,274){\sx{3.}{$x$}}
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| %
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| %\put(242,620){\sx{1.8}{\rot{85}$y\!=\!\mathcal{A}(4,x)\!=\! A_2(x\!+\!3)\!-\!3\!=\!\mathrm{tet}_2(x\!+\!3)\!-\!3$\ero}}
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| \put(248,720){\sx{1.8}{\rot{85}$y\!=\!\mathcal{A}(4,x)\!=\! A_{2,4}(x\!+\!3)\!-\!3$\ero}}
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| \put(312,720){\sx{1.8}{\rot{80}$y\!=\!\mathcal{A}(3,x)\!=\! A_{2,3}(x\!+\!3)\!-\!3$\ero}}
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| \put(348,712){\sx{1.8}{\rot{63}$y\!=\!\mathcal{A}(2,x)\!=\! A_{2,2}(x\!+\!3)\!-\!3$\ero}}
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| \put(314,526){\sx{1.8}{\rot{45}$y\!=\!\mathcal{A}(1,x)\!=\! A_{2,1}(x\!+\!3)\!-\!3$\ero}}
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| | |
| %\put(438,714){\sx{1.8}{\rot{85}$y\!=\!\mathrm{pen}(x)$\ero}}
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| %\put(538,912){\sx{1.8}{\rot{82}$y\!=\!\mathrm{tet}_2(x)$\ero}}
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| \put(526,822){\sx{1.8}{\rot{82}$y\!=\!A_{2,4}(x)\!=\!\mathrm{tet}_2(x)$\ero}}
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| %\put(578,892){\sx{1.8}{\rot{73}$y\!=\!2^x$\ero}}
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| \put(566,858){\sx{1.8}{\rot{73}$y\!=\!A_{2,3}(x)\!=\!2^x$\ero}}
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| \put(566,792){\sx{1.8}{\rot{62}$y\!=\!A_{2,2}(x)\!=\!\mathrm{2}x$\ero}}
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| %\put(478,628){\sx{1.8}{\rot{50}$y\!=\!\mathrm{e}\!+\!x$\ero}}
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| \put(520,696){\sx{1.96}{\rot{44}$y\!=\!A_{2,1}(x)\!=\!2\!+\!x$\ero}}
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| %
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| %\put(86,222){\sx{1.9}{\rot{11}$y\!=\!\mathrm{exp}(x)$\ero}}
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| %\put(20,30){\sx{1.9}{\rot{30}$y\!=\!\mathrm{pen}(x)$\ero}}
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| \put(32,326){\sx{1.9}{\rot{3}$y\!=\!2^x$\ero}}
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| \put(132,4){\sx{1.9}{\rot{81}$y\!=\!\mathrm{tet}_2(x)$\ero}}
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| \put(178,8){\sx{1.9}{\rot{66}$y\!=\!\mathrm{2} x$\ero}}
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| %
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| %\put(308, 13){\sx{2.2}{$y\!=\!L_{\mathrm e,4,0}$}}
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| \end{picture}
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| \end{document}
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| </nowiki>
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| | |
| ==Refrences==
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| http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html <br>
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| http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf <br>
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| http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf
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| D.Kouznetsov. (2009). Solution of F(z+1)=exp(F(z)) in the complex z-plane. [[Mathematics of Computation]], 78: 1647-1670.
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| https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
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| http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf <br>
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| http://mizugadro.mydns.jp/BOOK/202.pdf
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| Д.Кузнецов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014. (In Russian), DOI:10.1090/S0025-5718-09-02188-7. Page 257, Figure 18.8.
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| http://mizugadro.mydns.jp/t/index.php/File:Ack4c.jpg
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| D.Kouznetsov. Holomorphic ackermanns. 2015, in preparation.
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| [[Category:Book]]
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| [[Category:BookPlot]]
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| [[Category:Tetration]]
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| [[Category:Natural tetration]]
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| [[Category:Complex map]]
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| [[Category:AMS]]
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| [[Category:C++]]
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| [[Category:Latex]]
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| [[Category:TORI]]
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